integral of (4x^3-\sqrt[4]{x})/(2x)

解答

\int \frac{4 x ^{3} - 4 x}{2 x} d x

\frac{2}{3} (x^{3} - 3 4 x) + C

求解步骤

\int \frac{4 x ^{3} - 4 x}{2 x} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{4 x ^{3} - 4 x}{x} d x

乘开

\frac{4 x ^{3} - 4 x}{x} : 4 x^{2} - \frac{1}{x ^{\frac{3}{4}}}

= \frac{1}{2} \cdot \int 4 x^{2} - \frac{1}{x ^{\frac{3}{4}}} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int 4 x^{2} d x - \int \frac{1}{x ^{\frac{3}{4}}} d x)

\int 4 x^{2} d x = \frac{4 x ^{3}}{3}

\int \frac{1}{x ^{\frac{3}{4}}} d x = 4 x^{\frac{1}{4}}

= \frac{1}{2} (\frac{4 x ^{3}}{3} - 4 x^{\frac{1}{4}})

化简

\frac{1}{2} (\frac{4 x ^{3}}{3} - 4 x^{\frac{1}{4}}) : \frac{2}{3} (x^{3} - 3 4 x)

= \frac{2}{3} (x^{3} - 3 4 x)

解答补常数

= \frac{2}{3} (x^{3} - 3 4 x) + C

x \to - 2 lim (x^{2} + 2 t - 1)

\int (\frac{x + 1}{x})^{2} d x

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y^{^{'}} = 0.05 y - 48000

\int \frac{cos ( \frac{π}{x ^{12}} )}{x ^{13}} d x